<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">AFP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">p'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">q'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">AFP</font> <a href="aff_2.html#NR11" target="_self">satisfies_des</a> 
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E25</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:15"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E5:15"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E7:15"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">p'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">q'</font>
 <b>and </b><br/><a NAME="E8:15"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E9:15"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E10:15"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E11:15"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y'</font>
 <b>and </b><br/><a NAME="E12:15"/><i><font color="Green">E36</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E13:15"/><i><font color="Green">E37</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p'</font>
 <b>and </b><br/><a NAME="E14:15"/><i><font color="Green">E39</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">p'</font>
 ;<br/>

<a NAME="E15:15"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">p'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">q'</font>
 
<b>by </b><i>, , , , <a class="ref" href="translac.html#D1" target="_self">Def1</a>, <a class="txt" href="translac.html#E4:15"><i><font color="Green">E26</font></i></a>, <a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<a NAME="E16:15"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E17:15"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:15_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
contradiction
 <b>by </b><i><a class="txt" href="translac.html#E4:15"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:15"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font>
 ;<br/>

<a NAME="E2:15_2"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="translac.html#D1" target="_self">Def1</a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:15_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p'</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:15_2"/>
contradiction
 <b>by </b><i><a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:15"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_3"/><i><font color="Green">E72</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:15_3"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:15_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="txt" href="translac.html#E4:15"><i><font color="Green">E26</font></i></a>, <a class="txt" href="translac.html#E9:15"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E4:15_3"/>
( <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="txt" href="translac.html#E7:15"><i><font color="Green">E31</font></i></a>, <a class="txt" href="translac.html#E10:15"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:15_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:15_3"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:15_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E8:15_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">p</font> )
 
<b>by </b><i><a class="txt" href="translac.html#E11:15"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:15_3"/>
contradiction
 <b>by </b><i>, <a class="txt" href="translac.html#E4:15"><i><font color="Green">E26</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:15"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="txt" href="translac.html#E2"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="translac.html#E2:15"><i><font color="Green">E24</font></i></a>, <a class="txt" href="translac.html#E5:15"><i><font color="Green">E28</font></i></a>, <a class="txt" href="translac.html#E6:15"><i><font color="Green">E29</font></i></a>, <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>
<a NAME="E21:15"/><i><font color="Green">E76</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>,<font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_4"/><i><font color="Green">E77</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:15_4"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E3:15_4"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="translac.html#D1" target="_self">Def1</a>, <a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="translac.html#E11:15"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E4:15_4"/>
( <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, , <a class="txt" href="translac.html#E8:15"><i><font color="Green">E32</font></i></a>, <a class="txt" href="translac.html#E12:15"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:15_4"/><b>then </b>
<font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<a NAME="E6:15_4"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">p'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E7:15_4"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p'</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p'</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/>
<a NAME="E8:15_4"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p'</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p'</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">p'</font>,<font color="Maroon">p'</font> )
 
<b>by </b><i><a class="txt" href="translac.html#E13:15"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/>
<b>hence </b><a NAME="E9:15_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="translac.html#E3"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:15"/>
<font color="Maroon">p'</font>,<font color="Maroon">q'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="translac.html#E2"><i><font color="Green">Lemma22</font></i></a>, <a class="txt" href="translac.html#E4"><i><font color="Green">Lemma30</font></i></a>, <a class="txt" href="translac.html#E7:15"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/>
<b>hence </b><a NAME="E23:15"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="txt" href="translac.html#E1:15"><i><font color="Green">E23</font></i></a>, <a class="txt" href="translac.html#E3:15"><i><font color="Green">E25</font></i></a>, <a class="txt" href="translac.html#E9:15"><i><font color="Green">E33</font></i></a>, <a class="txt" href="translac.html#E10:15"><i><font color="Green">E34</font></i></a>, <a class="ref" href="transgeo.html#T97">TRANSGEO:97</a></i>;<br/>


</div>
